Is private research with low funds still feasible in the field of AI?
Yes, you can do research in Artificial Intelligence with low funds (but you need a lot of time!). Notably, because AI is not the same as applied machine learning (indeed running ML programs on big data requires a lot of computer power). For example, knowledge representation and reasoning or natural language processing (both are subfields of AI) generally don't require a lot of computer power. Even when interested in machine learning (which is not the same as AI, just a subfield of it among others), you can use a powerful PC or laptop for that.
A lot of recent papers (probably most of them) in journals like  Artificial Intelligence, or in conferences like IJCAI (see their past proceedings) are somehow theoretical, and when something is implemented, it runs on a laptop or desktop. Notice that both AI journal and IJCAI conferences are peer-reviewed with a very selective process.
Actually it is difficult and rare to find a research paper in AI mentioning that costly equipment was needed to do the research. Costly supercomputers used for research are generally not used by AI researchers (but by researchers in physics or bioinformatics), and AI researchers often don't even have access to such facilities.
For examples, recent IJCAI2016 papers such as Coco: Runtime Reasoning About Conflicting Commitments, Interdependent Scheduling Games, Control of Fair Division, Verifying Pushdown Multi-Agent Systems against Strategy Logics, etc. don't mention any costly computation.
Actually, it is likely that most recent papers don't use and don't need large scale costly cloud computing. And some of that research might not have been implemented (perhaps by some intern) in anything more than a toy prototype.



Please include in your answer examples of recent research results and the required infrastructure that was used to create them.


Here are some recent research publications, they all explicitly mention the needed computer equipment, follow the link to read them:
Event recent IJCAI experimental papers like A Multicore Tool for Constraint Solving, Compiling Constraint Networks
into Multivalued Decomposable Decision Graphs, External Memory Bidirectional Search, Multiple Constraint Acquisition, Completion of Disjunctive Logic Programs, Eliminating Disjunctions in Answer Set Programming by Restricted Unfolding and On the Empirical Time Complexity of Random 3-SAT at the Phase Transition mention at most a multi-core workstation (e.g. at most a dual Xeon socket workstation,  often a cheaper laptop or desktop). I guess that most authors need their GPU only as a graphics card, to use their display screen. BTW, papers in other journals, such as JAIR (like Improving the Efficiency of Dynamic Programming on Tree
Decompositions via Machine Learning), when they mention computer equipment, gives similar kind of machines (laptop or at most high-end desktop).
(Actually, I don't remember having read an AI paper mentioning costly computing equipment; and I believe the reason for that is that in the academic community access to supercomputers is nearly reserved to research in other domains: physics, bioinformatics, etc. For an AI researcher gaining such access is difficult and uncommon; BTW, in H2020 European research grants computing cost above 15% of the labor cost needs to be dully justified, so is exceptional)
However, you'll better publish your software as free software or open-source, and you need a lot of time (preferably full-time, or at least half-time) to do the research work, publish it, and follow outside progress in your area. BTW, contacting a nearby university could be helpful (you could attend some seminars, etc.)
You can find interesting blogs (e.g. the one of J.Pitrat) of AI researchers working on just a laptop or a desktop.
So, most research in AI (even in Machine Learning, read papers from JMLR) are done by researchers working on a laptop or desktop. I am not even sure you need a powerful GPU to do the research. You certainly can do interesting research with a desktop computer (probably running Linux, with 32Gb RAM, some AMD Ryzen or Intel i5 or i7) costing in 2017 less than 1500€ or maybe 2500€ (and sometimes a low end laptop is enough). In rare cases (a small minority of papers), you might need a dual-socket workstation (something costing perhaps 5000€).
Notice that Big Data is not considered as a field of Artificial Intelligence. I guess (but don't really know) that research in Big Data requires access to more computer power than a PC. BTW, I am not sure that AlphaGo is really a research project, it is more an industrial demo.
PS. See also http://norvig.com/21-days.html (notice who is the author!) and this Bismon draft report (developing programming techniques, in a GPLv3+ alpha-stage software prototype, notably frame-based or semantic-network related ones, with reflection and introspection, reusable and relevant to AI), and the RefPerSys project.